Editing Continuous wavelet transform (section)

==Definition==
The continuous wavelet transform of a function <math>x(t)</math> at a scale <math>a\in\mathbb{R^{+*}}</math> and translational value <math>b\in\mathbb{R}</math> is expressed by the following integral

:<math display="block">X_w(a,b)=\frac{1}{|a|^{1/2}} \int_{-\infty}^\infty x(t)\overline\psi\left(\frac{t-b}{a}\right)\,\mathrm{d}t</math>

where <math>\psi(t)</math>  is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of [[complex conjugate]]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal <math>x(t)</math>, the first inverse continuous wavelet transform can be exploited.

:<math>x(t)=C_\psi^{-1}\int_{0}^{\infty}\int_{-\infty}^{\infty} X_w(a,b)\frac{1}{|a|^{1/2}}\tilde\psi\left(\frac{t-b}{a}\right)\, \mathrm{d}b\ \frac{\mathrm{d}a}{a^2}</math>

<math>\tilde\psi(t)</math> is the [[Dual wavelet|dual function]] of <math>\psi(t)</math> and 
:<math>C_\psi=\int_{-\infty}^{\infty}\frac{\overline\hat{\psi}(\omega)\hat{\tilde\psi}(\omega)}{|\omega|}\, \mathrm{d}\omega</math>
is admissible constant, where hat means Fourier transform operator. Sometimes, <math>\tilde\psi(t)=\psi(t)</math>, then the admissible constant becomes
:<math>C_\psi = \int_{-\infty}^{+\infty}
  \frac{\left| \hat{\psi}(\omega) \right|^2}{\left| \omega \right|} \, \mathrm{d}\omega
</math>
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies 
:<math>0<C_\psi <\infty</math>
is called an admissible wavelet. To recover the original signal <math>x(t)</math>, the second inverse continuous wavelet transform can be exploited.
:<math>x(t)=\frac{1}{2\pi\overline\hat{\psi}(1)}\int_{0}^{\infty}\int_{-\infty}^{\infty} \frac{1}{a^2}X_w(a,b)\exp\left(i\frac{t-b}{a}\right)\,\mathrm{d}b\ \mathrm{d}a</math>
This inverse transform suggests that a wavelet should be defined as
:<math>\psi(t)=w(t)\exp(it) </math>
where <math>w(t)</math> is a window. Such defined wavelet can be called as an analyzing wavelet, because it admits to time-frequency analysis. An analyzing wavelet is unnecessary to be admissible.